Well not exactly - but Yes gba has Re-writable backup Flash
Cartridges that can be used to hold backup gba roms. Download
the game from the net for free - write it to this flash cards
and you don't need to buy the original cartridges any more.

GBA ROMS & Gameboy Advance Emulator

Sony PSP will have enough processing power to
mulate GAME BOY, GBC and even GBA roms but if you dont want to
wait for the realease of GBA emulator for PSP I suggest you downlod
a PC GBA emulator so that you can play the games on your Computer.
(Emu for Widows, Mac, Linux and BeOS available)

With gba emulator
you can play Gameboy, Gameboy Color and new Gameboy Advance (GBA SP) free game
roms on a PC or Mac. Download GBA emulator and emulator rom files
that contain the free gamez. Install the emu on pc and from within
the gba emulator window open the roms.

Gameboy Advance Emulator for PSP4.9 of 5 from 7531 reviews.

About GBA ROMS for Emulator

Gameboy Advace Roms (a.k.a. GBA ROMS) are the files with the
games. They are made using Gameboy Flash Linkers like
Flash Advance, Flash2Advance, FA Xtreme USB, XG-Flash, XG2, EZ-Flash, EZF Advance,
EZ-Flash II PowerStar, WiseBox and others. What flash linkers
/ writer do is they work as adapters so you can connect the Nintendo
cartridge to the PC USB or Printer port and backup
the game to the computer. On the compter the game is stored as
a *.gba file. Now these files can be used in the emulator or
you can take the same linker, put a flash card in it and write
the rom to this gba flash card. Multiple games can be stored
one one cuch cartridge!

Where to get gba roms?
Download gba roms from Gameoy-Advance-Roms.com -
They have a collection of over 1500 free gba roms and links
to websites with backup commercial game downloads.